Scalar field

In mathematics and physics, a scalar field associates a scalar to every point in space. Scalar fields are often used in physics, for instance to indicate the temperature distribution throughout space, or the air pressure.

Contents

1 Definition 2 Examples found in physics 3 Other kinds of fields 4 Difference between scalar and vector field 4.1 Example 1 4.2 Example 2 4.3 Visualization of coordinates 5 Differential geometry

Definition

A scalar field is a function from Rⁿ to R. That is, it is a function defined on the n-dimensional Euclidean space with real values. Often it is required to be continuous, or one or more times differentiable, that is, a function of class C^k.

The scalar field can be visualized as a n-dimensional space with a real or complex number attached to each point in the space.

The derivative of a scalar field results in a vector field called the gradient.

Examples found in physics

• Potential field
• In quantum field theory a scalar field is associated with spin 0 particles, like mesons. The scalar field may be real or complex valued (depending on whether it will associate a real or complex number to every point of space-time). Complex scalar fields represent charged particles.

Other kinds of fields

• Vector fields, which associate a vector to every point in space. Some examples of vector fields include the electromagnetic field and the Newtonian gravitational field.
• Tensor fields, which associate a tensor to every point in space. In general relativity, gravity is associated with a tensor field. In particular, with the Riemann curvature tensor. In Kaluza-Klein theory spacetime is extended to five dimensions and its Riemann curvature tensor can be separated out into ordinary four-dimensional gravitation plus an extra set, which is equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the "dilaton".

Difference between scalar and vector field

The difference between a scalar and vector field is not that a scalar is just one number while a vector is several numbers. The difference is in how their coordinates respond to coordinate transformations. A scalar is a coordinate whereas a vector can be described by coordinates, but it is not the collection of its coordinates.

Example 1

This example is about 2-dimensional Euclidean space (R²) where we examine Euclidean (x, y) and polar (r, θ) coordinates (which are undefined at the origin). Thus x = r cos θ and y = r sin θ and also r² = x² + y², cos θ = x/(x² + y²) and sin θ = y/(x² + y²). Suppose we have a scalar field and a vector field which are both given in polar coordinates by the constant function 1. More precisely by the functions

<math>s_{\mathrm{polar}}:(r, \theta) \mapsto 1, \quad v_{\mathrm{polar}}:(r, \theta) \mapsto (1, 0).<math>

Thus we have a scalar field which has the value 1 everywhere and a vector field which attaches a vector in the r-direction with length 1 to each point. Now what Euclidean coordinates describe the same fields? A vector in the r-direction at a point (a, b)_Euclidean points is cos θ(a, b) in the x-direction and sin θ(a, b) in the y-direction. Thus in Euclidean coordinates the same fields are described by the functions

<math>s_{\mathrm{Euclidean}}:(x, y) \mapsto 1, \quad v_{\mathrm{Euclidean}}:(x, y) \mapsto (\cos \theta(x, y), \sin \theta(x, y)) = (\frac{x}{x^2 + y^2}, \frac{y}{x^2 + y^2}).<math>

These functions aren't even close, but you might still think that the number of coordinates is also different, so we need another example.

Example 2

To completely dispell any residual belief in number of coordinates we will now work in 1-dimensional Euclidean space (R) with its standard Euclidean coordinate x. We will have to use some unusual coordinate however, namely ξ := 2x. Suppose we have a scalar field and a vector field which are both given in unusual coordinates by the constant function 1. More precisely by the functions

<math>s_{\mathrm{unusual}}:\xi \mapsto 1, \quad v_{\mathrm{unusual}}:\xi \mapsto 1.<math>

Thus we have a scalar field which has the value 1 everywhere and a vector field which attaches a vector in the ξ-direction with magnitude 1 unit of ξ to each point. But if ξ changes one unit then x changes 2 units. Thus this vector has a magnitude of 2 in units of x. Thus these fields in Euclidean coordinate are described by the functions

<math>s_{\mathrm{Euclidean}}:x \mapsto 1, \quad v_{\mathrm{Euclidean}}:x \mapsto 2,<math>

which are different.

Visualization of coordinates

Basically all you need to know about two different coordinate systems is how to get from one to the other, but it usually helps to visualize one as a rectangular grid. But you can also do that with polar coordinates. How weird Euclidean coordinates are when you do this. If there is more structure, such as a metric, then you know about the angles between your coordinate vectors and their lengths, and in that way you can choose which ones are more rectangular.

Differential geometry

A scalar field on a C^k-manifold is a C^k function to the real numbers. Taking Rⁿ as manifold gives back the special case of vector calculus.

A scalar field is also a 0-form. See differential forms.de:Skalarfeld fr:Champ scalaire sv:Skalärfält zh:标量场 pl:Pole skalarne